Sure (well kinda...), but why does it have to be continuous, rather than discrete, to progress?
Well, consider the real numbers between 0 and 1, inclusive. They're continuous, in that there are no "gaps" between numbers (to be more precise they're topologically connected), but they have a start and end.
I don't know what you mean by "pre-time"; note that if time started with the Big Bang then it's not really meaningful to talk about "before", since that suggests an earlier time.
Why do you say that? It surely didn't.
Well if it makes you feel any better, nobody completely understands time. We know what role it plays in our theories of physics, and those theories seem to work, but AFAIK there's no definite answer to the question of why we experience time the way we do, namely as something that progresses. This may turn out to be explained by the working of our brains or it may turn out that some fundamental component is missing from our understanding of physics.